Concolor

It’s pleasant that people celebrate my birthday with fireworks.  Actually they’re celebrating the birthday of the U.S. of A.  This July 4, Mercury will contribute a small firework by coming to the peak of its excursion into the morning sky.

See the end note about enlarging illustrations.

That is, it will be at an elongation of 21.5° westward from the Sun.

It’s a pity there doesn’t happen to be a meteor shower painting streaks across the July 4 sky.  Well, it isn’t difficult to invent one.

The timing of Mercury’s maximum elongation is 20h Universal Time, which is 5 or more hours earlier by American clocks, but that hardly matters – it is a soft event, like the top of a rounded hill.

Mercury is now shining at magnitude 0.5, a bit dimmer than the star Capella and brighter than Aldebaran.  But it is, at the time and American location chosen for the picture, only about 4° above the horizon.  And as the broad arrow shows, in an hour the little planet will be higher but the Sun will be up.

You may glimpse Jupiter and the waning Moon, but don’t expect to see, in the bright twilight, all the picture’s other features.  They orient us to the direction in which we’re looking – ahead along Earth’s orbit, and out toward the nearest edge of our Galaxy – and you might have seen some of them if you were out earlier, waiting for Mercury to rise.

The arrows through the planets and the Sun show their movement, in relation to the starry background, over 5 days.  (Two days ahead, Mercury will be near Zeta Tauri, the star marking the tip of the Bull’s southern horn.)  At the date of westernmost elongation, Mercury starts to move eastward slightly faster than the Sun, toward its passage behind the Sun on August 1.

We summarized in January Mercury’s three morning and three-and-a-half evening appearances of 2021.

As this graph shows, the elongation (the filled curves, blue for morning and gray for evening apparitions) reaches higher than the actual altitude the planet attains above the sunset (or sunrise) horizon.

For latitude 40° north (the thick lines), the best morning will be July 9, and the maximum sunset altitude only 14.2°.  By contrast, for a south-hemisphere latitude such as 35° (thin lines), the situation is better: the sunset altitude reached 16.9° on July 1.  For both hemispheres, this is a middling-quality Mercury performance.

As you can see in the sky scene, Mercury is at present traveling south of the ecliptic, which is why it is more favorably placed for south-hemisphere viewers.  More generally, the reason for all these asymmetries is Mercury’s orbit, more elliptical than those of the other major planets and more inclined to the ecliptic (the plane of Earth’s orbit).

Con colores

Old Glory and the Union Jack, the flags of the US and the UK, are red-white-and-blue, so let’s turn the sky into a flag.

Over more than a week I’ve been doing work (whose intricacy I’ll try not to bore you with) to make the part of my programming that draws constellations both simpler and more powerful.

Since 1930, there have been 88 official constellations (one of them, Serpens, divided into two non-contiguous parts).  Their boundaries consist of north-south and east-west lines, approximating the traditional areas.  Benjamin Gould determined many of these points in 1875 and later Eugène Delporte completed them for the southern sky.

There are 784 of these line segments, and the catalog of them goes like this:

22h52   34 30   52 30  AndLac
52 30   22h52   23h20  AndCas
23h20   52 30   50 00  AndCas
50 00   23h20   23h35  AndCas

The first line is part of the boundary between Andromeda and Lacerta (the small “lizard” constellation), and is along right ascension 22h52m from declination 34°30′ to 52°30′, so it is “vertical.”  The alternate lines are “horizontal,” along parallels of declination.

It’s relatively easy to calculate the positions of these points.  You convert them to decimal degrees; then, since they are for Gould’s epoch of 1875, you have to correct them for precession (the slow shift of the entire map of the sky caused by Earth’s changing tilt).  Also, each segment is not really a straight line: it curves, depending on the projection of the particular map, so I have to make my program calculate not just for the corner points but for a number of points along each segment.

So we can draw all these boundaries, or those of selected constellations, as lines.

But what if I want to make a constellation area into an “object” (as computerese calls it) that can be filled with a color?  This is hazardous.

In the way I’ve taught myself to program with Fortran, the enormous sequences of formulae end with instructions in the language used by Adobe Illustrator (a language I had to discover surreptitiously by what I think is called “back-programming”).  For a closed object, there has to be a certain code for the first point of the object and a certain code for the closing point.  If one of them is missing or in the wrong place, the program bombs.  This can happen because of the complexity of the program or if part of an object is not drawn.  As with the constellations around the edges of the picture.

To make it easier to handle one constellation at a time, I restructured the Gould-Delporte list of segments into a less severely compact form –

34 30 22h52  | 52 30 22h52  |And
52 30 22h52  | 52 30 23h20  |And

– in which each constellation has its complete list with its segments in order,  every segment having to be listed twice.

Now we can play with colors.  My mosaic sky dome is for latitude 40° north at sidereal time 5h, such as on a January evening.

In white are the constellations of the zodiac (selected by a “Z” in the instruction file that the program reads).  In red are all the non-ancient ones, those not in Ptolemy’s Almagest of about 140 AD (so I code them “-P”).  In blue are all the rest (“+”).

And single constellations can be selected with code “1,” so I’ve shown dragon Draco and made him yellow.

I could select all the non-zodiacal ones (“-Z”), or all the ancient (“P”).  And perhaps you would like me to devise other categories, such as southern (“-N”) or circumpolar (“C”).  Or (“!” and “-!”) the most conspicuous (such as Orion) and least (such as Lynx).  Or the constellations that were made by breaking up the ancient Argo (“A”).

Names

The placing of them is another problem.

The computer runs through thousands of operations to produce an illustration in microseconds that seem like an instant.  But what I call “dressing” the illustration – using the mouse to adjust the labels – can take an hour.  So anything that better automates their placement is welcome.

I’ve been using a file of positions, so that for instance “Orion” is conveniently just above the stars of his belt and doesn’t overlie them.  But if a chart is of a small region, for instance showing this year’s movements of Jupiter, part of a constellation may appear but not its name.  It can be pasted in by some other method, but that takes even more time.

Now, for any constellation that gets into the picture, I can use all the points around its boundary, or all that get into the picture, to find a central position for the label.  And this is how I’ve let them appear in that picture.

But there are at least two ways of doing it.  One is to use the point that is median between the boundary’s northernmost, southernmost, easternmost, and westernmost points.  The other is to take the average of all the points.

You can see why the first way doesn’t work perfectly if you look at yellow Draco.  That monster is so necky that his median point is not where we would think of as his center.  And the averaging method gives more weight to the parts of a constellation’s boundary that consist of many short steps, such as the northern boundary of Monoceros.  I may try averaging the two methods, or finding some other.

By “in the picture,” I’m including a certain margin beyond the picture’s edge, so that some further needed labels are available.  Aquarius, over on the right next to Aries – its name doesn’t appear.  And next to Pyxis is a small fraction of another constellation, Antlia.  These labels are hidden under the gray mask around the picture, and I could get them from there and move them into view.  There may be a better way around this.

A worse problem would be revealed if the picture included Pisces, a constellation that straddles the zero line of right ascension.  So some points in its boundary have right ascensions of not much more than zero, and the others are not much below 360°, with a great gap between.  Neither their average nor their median makes sense.  Here is what happens if I try to map the whole sky and give Pisces a color.

The real Pisces consists of the small white bits at left and right and the north-projecting red bit at right.

This, which I might call the Cross-Zero Problem or the Pisces Problem, has wrung much thought out of my non-mathematical brain, and needs more.

– Concolor really means “uniformly colored.”  The cougar is Puma concolor, and I remember that I once catalogued a book by or about a Mexican or other Latin American with a name something like Felix “Concolor” Bustamante.

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ILLUSTRATIONS in these posts are made with precision but have to be inserted in another format.  You may be able to enlarge them on your monitor.  One way: right-click, and choose “View image”, then enlarge.  Or choose “Copy image”, then put it on your desktop, then open it.  On an iPad or phone, use the finger gesture that enlarges (spreading with two fingers, or tapping and dragging with three fingers).  Other methods have been suggested, such as dragging the image to the desktop and opening it in other ways.

Sometimes I make improvements or corrections to a post after publishing  it.  If you click on the title, rather than on ‘Read more’, I think you are sure to see the latest version.

This weblog maintains its right to be about astronomy or anything under the sun.